You may start to work with your midterm exam at 8:00 am Friday October 7th, and you must finish it and submit your report and MATLAB code online before Monday October 17th 11:59 pm.

Problem 1. (25 points)

Write a MATLAB live code to sort a general M M (M is an odd integer) matrix in an ascending order, and then re-arrange it in a ascending spiral pattern such that the smallest element is in the center of the matrix (you may have multiple entries that have the same smallest value, but one of them is in the middle), and the largest elements are in the boundary of the matrix. For example, if you are given the following matrix (test-case example),

15 2 8 12    10

11 9 3 14 5

6 4 1 7 13

7 11    14 5 7

1 2 4 5 6

your code must be able to sort the matrix in ascending order in terms of row

1 1 2 2 3

6 6 7 7 7

8 9 10     11    11

12    13    14    14    15

and then re-arrange it into the following spiral pattern ascending from the mid- dle in counter-clock wise direction

  .

 

Then  generate a 11 11 matrix A = randi(100, 11), and sort and re-arrange it into the spiral pattern and visualize it by using the MATLAB command:

surf (A).

Note: Use of built-in search (find) or sort functions are prohibited.

Problem 2. (25 points)

Consider the piece-wise continuous function

. x² ,   x < 0

(1) Plot the function in the range [ 3, 3];

(2) Integrate the function in the interval [-3, 3] analytically, i.e.

∫ ₃

(3) Write  a MATLAB  code to integrate the function from 3 to 3 by using the Riemann sum, i.e.

n

R = f (x^∗i )∆x_i

i=1

where  a = x₀ < x₁ < x₂ < · · · < x_n = b,  and  x^∗i  ∈ [x_i−1, x_i]  (Note  that  there n+1 points here, and ∆x_i = (b − a)/n).  Choose (I) x^∗i  = x_i; (II) x^∗i  = x_i−1, and

(III) x^∗i  =  ¹ (x_i−1 + x_i).

(4) Choose a = 3 and b = 3. Calculate R and compare the value with the

exact solution, i.e. calculate the relative numerical integration error as follows,

e = R − Rexact

Rexact

where you are asked to choose n = 100 and n = 1000, and

(4) In your midterm report, you must present the numerical integration errors

e for the above three different cases, i.e.,

(1)x^∗i  = x_i; (2)x^∗i  = x_i−1, and (3)x^∗i  =  ¹ (x_i−1 + x_i).

Discuss your findings in your report.

Problem 3. (25 points)

Write a general binary search code that can find a target with multiple appear- ances in an array. You can only use binary search algorithm. Use the following array to test your code to see if your code works:

A = [2, 4, 0, 1, 1, 2, 3, − 3, 4, − 4, 4, 5, 2, 8, 1, 10, 10, 1];

and choose the test targets as 1. Your code must return the indices of the target number in the original un-sorted array.

In your MATLAB code, no MATLAB built-in search or sort functions are allowed to be used.

Problem 4. (25 points) [Collatz problem] Consider the following function,

f (n) =



n

, if n is even;

2

3n + 1, if n is odd; where n is a natural number or positive integer.

Now write a MATLAB recursive function by performing the following oper- ation repeatedly, beginning with any positive integer, n, and taking the result at each step as the input for the next step. That is

a_i =

 f (a_i−1), i = 1, 2, 3, · · · k